PRELIMINARY EXAMINATION: APPLIED MATHEMATICS I
January 7, 2013, 1:00–2:30 p.m.
Work all 3 of the following 3 problems.
1. Let H be a real Hilbert space and suppose that P is a bounded linear projection on H. Let
Q = I − P and define M = P (H) and N = Q(H).
(a) Show that there exists C > 0 such that
kx − P xk ≤ C inf kx − yk
for all x ∈ H.
y∈M
(b) Prove that P is an orthogonal projection if and only if
inf
ky − xk = 1.
y∈N,kyk=1
x∈M
[Hint: For the converse, it is enough to show that for any z ∈ H, z − P z = Qz ⊥ M . Consider
y = Qz/kQzk.]
2. Let X and Y be separable and reflexive Banach spaces and let T ∈ B(X, Y ). Suppose that
w
yn ∈ Y and yn * y (yn converges weakly to y), and that there are xn ∈ X such that
T xn = yn
for all n.
Moreover, suppose that kxn k ≤ M for some M > 0.
(a) State the Generalized Heine-Borel Theorem for a separable and reflexive Banach space.
w
(b) Show that for some subsequence {xnk }∞
k=1 , T xnk * y.
(c) If T is a compact operator, show that the convergence in (b) is strong for the entire subsequence {xnk }∞
k=1 .
3. Let H be a complex Hilbert space and T ∈ C(H, H) a compact linear operator.
(a) Show that T ∗ T is compact, self-adjoint, and positive (i.e., nonnegative) on H.
(b) Show that if w ∈ N (T ∗ T ), then in fact T w = 0.
∞
(c) Show that there are a pair of orthonormal sets {un }∞
n=1 and {vn }n=1 such that for any
x ∈ H,
∞
X
Tx =
λn hx, un i vn ,
n=1
{λ2n }∞
n=1
where
vectors.
are the positive eigenvalues of T ∗ T and {un }∞
n=1 are the corresponding eigen-